承上篇，須知『線性系統』除了有

Zero state response

In electrical circuit theory, the zero state response (ZSR), also known as the forced response is the behavior or response of a circuit with initial state of zero. The ZSR results only from the external inputs or driving functions of the circuit and not from the initial state. The ZSR is also called the forced or driven response of the circuit.

The total response of the circuit is the superposition of the ZSR and the ZIR, or Zero Input Response. The ZIR results only from the initial state of the circuit and not from any external drive. The ZIR is also called the natural response, and the resonant frequencies of the ZIR are called the natural frequencies. Given a description of a system in the s-domain, the zero-state response can be described as Y(s)=Init(s)/a(s) where a(s) and Init(s) are system-specific.

之外，還有 ZIR Zero Input Response 哩！故爾推論不夠嚴謹也。

想那『因果性』是說︰

系統之輸出不依賴『未來』的輸入。

和系統線性與否實不必相涉呦！！

或許一例可解惑耶？

Zero state response and zero input response in integrator and differentiator circuits

One example of zero state response being used is in integrator and differentiator circuits. By examining a simple integrator circuit it can be demonstrated that when a function is put into a linear time-invariant (LTI) system, an output can be characterized by asuperposition or sum of the Zero Input Response and the zero state response.

A system can be represented as

$\displaystyle f(t)$ $\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}f(\tau )d\tau$

with the input $\displaystyle f(t).$ on the left and the output $\displaystyle y(t).$ on the right.

The output $\displaystyle y(t).$ can be separated into a zero input and a zero state solution with

$\displaystyle y(t)=\underbrace {y(t_{0})} _{Zero-input\ response}+\underbrace {\int _{t_{0}}^{t}f(\tau )d\tau } _{Zero-state\ response}.$

The contributions of $\displaystyle y(t_{0})$ and $\displaystyle f(t)$ to output $\displaystyle y(t)$ are additive and each contribution $\displaystyle y(t_{0})$ and $\displaystyle \int _{t_{0}}^{t}f(\tau )d\tau$ vanishes with vanishing $\displaystyle y(t_{0})$ and $\displaystyle f(t).$

This behavior constitutes a linear system. A linear system has an output that is a sum of distinct zero-input and zero-state components, each varying linearly, with the initial state of the system and the input of the system respectively.

The zero input response and zero state response are independent of each other and therefore each component can be computed independently of the other.

Zero state response in integrator and differentiator circuits

The Zero State Response $\displaystyle \int _{t_{0}}^{t}f(\tau )d\tau$ represents the system output $\displaystyle y(t)$ when $\displaystyle y(t_{0})=0.$

When there is no influence from internal voltages or currents due to previously charged components

$\displaystyle y(t)=\int _{t_{0}}^{t}f(\tau )d\tau .$

Zero state response varies with the system input and under zero-state conditions we could say that two independent inputs results in two independent outputs:

$\displaystyle f_{1}(t)$ $\displaystyle y_{1}(t)$

and

$\displaystyle f_{2}(t)$ $\displaystyle y_{2}(t).$

Because of linearity we can then apply the principles of superposition to achieve

$\displaystyle Kf_{1}(t)+Kf_{2}(t)$ $\displaystyle Ky_{1}(t)+Ky_{2}(t).$

Verifications of zero state response in integrator and differentiator circuits

To arrive at general equation

Simple Integrator Circuit

The circuit to the right acts as a simple integrator circuit and will be used to verify the equation $\displaystyle y(t)=\int _{t_{0}}^{t}f(\tau )d\tau$ as the zero state response of an integrator circuit.

Capacitors have the current-voltage relation $\displaystyle i(t)=C{\frac {dv}{dt}}$ where C is the capacitance, measured in farads, of the capacitor.

By manipulating the above equation the capacitor can be shown to effectively integrate the current through it. The resulting equation also demonstrates the zero state and zero input responses to the integrator circuit.

First, by integrating both sides of the above equation

$\displaystyle \int _{a}^{b}i(t)dt=\int _{a}^{b}C{\frac {dv}{dt}}dt.$

Second, by integrating the right side

$\displaystyle \int _{a}^{b}i(t)dt=C[v(b)-v(a)].$

Third, distribute and subtract $\displaystyle Cv(a)$ to get

$\displaystyle Cv(b)=Cv(a)+\int _{a}^{b}i(t)dt.$

Fourth, divide by $\displaystyle C$ to achieve

$\displaystyle v(b)=v(a)+{\frac {1}{C}}\int _{a}^{b}i(t)dt.$

By substituting $\displaystyle t$ for $\displaystyle b$ and $\displaystyle t_{o}$ for $\displaystyle a$ and by using the dummy variable $\displaystyle \tau$ as the variable of integration the general equation

$\displaystyle v(t)=v(t_{0})+{\frac {1}{C}}\int _{t_{0}}^{t}i(\tau )d\tau$

is found.

To arrive at circuit specific example

The general equation can then be used to further demonstrate this verification by using the conditions of the simple integrator circuit above.

By using the capacitance of 1 farad as shown in the integrator circuit above

$\displaystyle v(t)=v(t_{0})+\int _{t_{0}}^{t}i(\tau )d\tau ,$

which is the equation containing the zero input and zero state response seen at the top of the page.

To verify zero state linearity

To verify its zero state linearity set the voltage around the capacitor at time 0 equal to 0, or $\displaystyle v(t_{0})=0$ , meaning that there is no initial voltage. This eliminates the first term forming the equation

$\displaystyle v(t)=\int _{t_{0}}^{t}i(\tau )d\tau$ .

In accordance with the methods of linear time-invariant systems, by putting two different inputs into the integrator circuit, $\displaystyle i_{1}(t)$ and $\displaystyle i_{2}(t)$ , the two different outputs

$\displaystyle v_{1}(t)=\int _{t_{0}}^{t}i_{1}(\tau )d\tau$

and

$\displaystyle v_{2}(t)=\int _{t_{0}}^{t}i_{2}(\tau )d\tau$

are found respectively.

By using the superposition principle the inputs $\displaystyle i_{1}(t)$ and $\displaystyle i_{2}(t)$ can be combined to get a new input

$\displaystyle i_{3}(t)=K_{1}i_{1}(t)+K_{2}i_{2}(t)$

and a new output

$\displaystyle v_{3}(t)=\int _{t_{0}}^{t}(K_{1}i_{1}(\tau )+K_{2}i_{2}(\tau ))d\tau .$

By integrating the right side of

$\displaystyle v_{3}(t)=K_{1}\int _{t_{0}}^{t}i_{1}(\tau )d\tau +K_{2}\int _{t_{0}}^{t}i_{2}(\tau )d\tau ,$

$\displaystyle v_{3}(t)=K_{1}v_{1}(t)+K_{2}v_{2}(t)$

is found, which implies the system is linear at zero state, $\displaystyle v(t_{0})=0$ .

This entire verification example could also have been done with a voltage source in place of the current source and an inductor in place of the capacitor. We would have then been solving for a current instead of a voltage.

如果此時重讀『線性系統』

Linear system

A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

Definition

A general deterministic system can be described by an operator, $\displaystyle H$ , that maps an input, $\displaystyle x(t)$ , as a function of $\displaystyle t$ to an output, $\displaystyle y(t)$ , a type of black box description. Linear systems satisfy the property of superposition. Given two valid inputs

\displaystyle x_{1}(t)

\displaystyle x_{2}(t)

as well as their respective outputs

\displaystyle y_{1}(t)=H\left\{x_{1}(t)\right\}

\displaystyle y_{2}(t)=H\left\{x_{2}(t)\right\}

then a linear system must satisfy

\displaystyle \alpha y_{1}(t)+\beta y_{2}(t)=H\left\{\alpha x_{1}(t)+\beta x_{2}(t)\right\}

for any scalar values $\displaystyle \alpha$ and $\displaystyle \beta$ .

The system is then defined by the equation $\displaystyle H(x(t))=y(t)$ , where $\displaystyle y(t)$ is some arbitrary function of time, and $\displaystyle x(t)$ is the system state. Given $\displaystyle y(t)$ and $\displaystyle H$ , $\displaystyle x(t)$ can be solved for.

For example, a simple harmonic oscillator obeys the differential equation:

\displaystyle m{\frac {d^{2}(x)}{dt^{2}} = - k x

\displaystyle H(x(t))=m{\frac {d^{2}(x(t))}{dt^{2}}}+kx(t)

then $\displaystyle H$ is a linear operator. Letting $\displaystyle y(t)=0$ , we can rewrite the differential equation as $\displaystyle H(x(t))=y(t)$ , which shows that a simple harmonic oscillator is a linear system.

The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems. For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function $\displaystyle x(t)$ in terms of unit impulses or frequency components.

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense.

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

Time-varying impulse response

The time-varying impulse response h(t₂,t₁) of a linear system is defined as the response of the system at time t = t₂ to a single impulse applied at time t = t₁. In other words, if the input x(t) to a linear system is

\displaystyle x(t)=\delta (t-t_{1})

where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is

\displaystyle y(t)|_{t=t_{2}}=h(t_{2},t_{1})

then the function h(t₂,t₁) is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied:

\displaystyle h(t_{2},t_{1})=0,t_{2}<t_{1}

The convolution integral

The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition:

\displaystyle y(t)=\int _{-\infty }^{t}h(t,t')x(t')dt'=\int _{-\infty }^{\infty }h(t,t')x(t')dt'

If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and h() is a function only of the time difference τ = t-t’ which is zero for τ<0 (namely t<t’). By redefinition of h() it is then possible to write the input-output relation equivalently in any of the ways,

\displaystyle y(t)=\int _{-\infty }^{t}h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(t-t')x(t')dt'=\int _{-\infty }^{\infty }h(\tau )x(t-\tau )d\tau =\int _{0}^{\infty }h(\tau )x(t-\tau )d\tau

Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is:

\displaystyle H(s)=\int _{0}^{\infty }h(t)e^{-st}\,dt.

In applications this is usually a rational algebraic function of s. Because h(t) is zero for negative t, the integral may equally be written over the doubly infinite range and putting s = iω follows the formula for the frequency response function:

\displaystyle H(i\omega )=\int _{-\infty }^{\infty }h(t)e^{-i\omega t}dt

文本，思考『因果關係』如何引入，自能體會乎？？

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每日彙整: 2018-07-24

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